I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.

(I am trying to find whether the map is area preserving? I know "the map $f:\mathbb{R}^n\to\mathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $\pm1$".)

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    $\begingroup$ What is the definition of Jacobian? $\endgroup$ – zoidberg Dec 9 '18 at 2:39

The Jacobian of a multivariable function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:

$$\textbf{J}(f(x,y)) = det\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{pmatrix}$$

where $det$ denotes the determinant of the matrix.

Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:

$$det\begin{pmatrix} 1 & 2y \\ 2x & 1 \\ \end{pmatrix}=1-4xy$$


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